Complex Indefinite Integrals

Complex Indefinite Integrals

Just like with real indefinite integrals, if $\frac{d}{dz} F(z) = f(z)$, we say $F(z)$ is an antiderivative of $f(z)$. All other antiderivatives of $f(z)$ are given by $F(z) + c$, $c \in \mathbb{Z}$ and we say the indefinite integral of $f(z)$ with respect to $dz$ is

\[\int{f(z)}dz = F(z) + c.\]

It seems that complex indefinite integration is much like real indefinite integration. At the very least, it is a linear operator so we homogeniety and additivity, and we can use the chain rule. We do need to be careful with where the integral is valid, and consider multivalued functions like $logz$ carefully in this regard.